Why is Andromache called Hector's "precious wife"? Mention three things that would make a woman precious.
Hector's own words in Book VI demonstrate how precious Andromache is to him:
And yet no dire presage so wounds my mind, My mother's death, the ruin of my kind,  Not Priam's hoary hairs defiled with gore, Not all my brothers gasping on the shore; As thine, Andromache! Thy griefs I dread: I see thee trembling, weeping, captive led!
Here, he expresses his dread fear that something terrible could happen to her in the event of his death, such as being taken prisoner (which is indeed her fate). In his mind, leaving her to her fate disturbs him more than losing his mother, his father, Priam, or his Trojan brothers. Implicit in this quotation is that their marriage was more than strategic; Hector truly loved Andromache.
In return, Andromache expresses her love and devotion to Hector, when in Book VI she expresses:
"Yet while my Hector still survives, I see My father, mother, brethren, all, in thee: Alas! my parents, brothers, kindred, all Once more will perish, if my Hector fall..."
It is clear that she puts her marriage and love for Hector ahead of her love for the rest of her family. To have a wife that is more than a function of a political marriage and the producer of an heir makes her precious to Hector.
Lastly, in this male-dominated cultural context, women were often the spoils of war; one needs only to consider the importance of Helen, the wife of Sparta's king, Menelaus, whose abduction sparked the Trojan War. Women were chattel, and in this way Andromache was precious, not just to her husband, but also to his enemies. After the Trojan war and Hector's death, Andromache was given to Neoptolemus as his concubine, and she gave birth to his son. Women in this culture were objectified, and Andromache was no exception to this truth, but Homer does convince the reader that her value to Hector was more than that of chattel.
She is precious to Hector because he loves her. She is the mother of his son who is still just a baby. She is not like the other women who are hiding in their houses or praying at the temple to "the awful goddess Minerva". She is out on the wall watching for her husband and the events of the battle.
She knows what Achilles is capable of doing since it was Achilles who killed her own father in the battle with the Cilicians. Andromache begs Hector to stay in the city with her and lead the battle from the safety of the walls. He attempts to comfort her by telling her that when his time comes, no one can stop it, but he would die just the same.
This scene in the Iliad serves to humanize Hector as we see the joy he takes in seeing his son Astyanax and his wife. Their love is elevated to the same level as the rage that pervades the epic. At this moment in the epic of the Iliad, nothing can touch the genuine love that this family represents. It is for this, the ideal love and family, that Hector must go out and fight.